Hebrew, then English. Open "What it means" under any line for the explanation.
After the ten psalms and the closing verses:
רִבּוֹנוֹ שֶׁל עוֹלָם, עִלַּת הָעִלּוֹת וְסִבַּת כָּל הַסִּבּוֹת,
Master of the world, cause of causes and reason behind all reasons,
אַנְתְּ לְעֵלָּא לְעֵלָּא מִן כֹּלָא, וְלֵית לְעֵלָּא מִנָּךְ, דְּלֵית מַחֲשָׁבָה תְּפִיסָא בָךְ כְּלָל,
You are above, high above everything, and nothing is above You, for no thought can take hold of You at all
What it means
You are high above everything, and nothing is above You, for no thought can take hold of You at all.
וּלְךָ דֻמִיָּה תְהִלָּה, וּמְרוֹמַם עַל כָּל בְּרָכָה וּתְהִלָּה.
And for You, silence is praise, and You are raised above every blessing and praise.
What it means
"For You, silence is praise" is Tehillim 65:2. Rashi (the main medieval commentator): there is no end to Your praise, so the more you add, the more you take away. "Raised above every blessing and praise" is Nechemiah 9:5, said every morning in Vayevarech David. The Halleluyah prayer (Tehillim 150) tells the Gemara's story about piling up praise.
Tehillim 65:2, with Rashi; Nechemiah 9:5; Siddur Sefard, Vayevarech Davidאוֹתְךָ אֶדְרֹשׁ אוֹתְךָ אֲבַקֵּשׁ, שֶׁתַּחְתֹּר חֲתִירָה דֶּרֶךְ כְּבוּשָׁה מֵאִתְּךָ,
You I seek, You I ask for: that You dig a tunnel, a well-worn road from You,
What it means
You I seek, You I ask for: that You dig a tunnel, a well-worn road, out from You,
Both verbs are in one verse: "From there you will seek Hashem your God, and you will find Him, when you search for Him with all your heart and all your soul" (Devarim 4:29). The verse comes right after a warning that Israel would be scattered among the nations, so "from there" means from wherever you ended up.
Devarim 4:27-29דֶּרֶךְ כָּל הָעוֹלָמוֹת, עַד הַהִשְׁתַּלְשְׁלוּת שֶׁלִּי בַּמָּקוֹם שֶׁאֲנִי עוֹמֵד,
by way of all the worlds, all the way to my own link in the chain, in the place where I am standing,
What it means
by way of all the worlds, all the way down to my own link in the chain, in the place where I am standing,
Hishtalshelut comes from shalshelet, a chain. Rabbi Natan of Breslov (the main student of Rebbe Nachman of Breslov, a Chassidic master who died in 1810) writes that the further the worlds came down that chain, away from the start of creation, the more it counts when they serve God: "the main honor comes from those who are far."
Rabbi Natan, Likutei Halakhot, Yoreh De'ah, Laws of Honoring Parents 2:2כְּפִי אֲשֶׁר נִגְלָה לְךָ יֹדֵעַ תַּעֲלֻמוֹת,
as it is revealed to You, Knower of hidden things,
What it means
"Knows hidden things" is from Tehillim 44:22: "He knows the secrets of the heart." In that psalm a people in trouble says: all this has come on us, and we did not forget You. Radak (Rabbi David Kimchi, Provence, about 1200) reads it as: if any thought of giving up our faith in Him had entered our hearts, God would know.
Tehillim 44:18-22, with Radakוּבַדֶּרֶךְ וּנְתִיב הַזֶּה תָּאִיר עָלַי אוֹרְךָ,
and on this road and path shine Your light on me,
What it means
A verse that asks for the same thing is Tehillim 43:3: "Send out Your light and Your truth; they will lead me." Radak says the light is the rescue itself. Rashi reads the light as the Mashiach (the king from David's family who will redeem Israel) and the truth as Eliyahu the prophet.
Tehillim 43:3, with Radak and Rashiלְהַחֲזִירֵנִי בִּתְשׁוּבָה שְׁלֵמָה לְפָנֶיךָ בֶּאֱמֶת, כְּפִי רְצוֹנְךָ בֶּאֱמֶת, כְּפִי רְצוֹן מִבְחַר הַבְּרוּאִים,
to bring me back in complete return before You, in truth, as You want, in truth, as the choicest of creatures want,
What it means
The Hashiveinu prayer gives the test of the Rambam (Maimonides, the great medieval Jewish thinker and legal authority, died 1204) for complete teshuvah: the same chance comes again, you could do it again, and you hold back because of your teshuvah, not out of fear or weakness.
Rambam, Mishneh Torah, Laws of Repentance 2:1לִבְלִי לַחֲשֹׁב בְּמַחֲשַׁבְתִּי שׁוּם מַחֲשֶׁבֶת חוּץ, וְשׁוּם מַחֲשָׁבָה וּבִלְבּוּל שֶׁהוּא נֶגֶד רְצוֹנֶךָ,
so that I do not think in my mind any outside thought, or any thought or confusion that goes against Your will,
רַק לִדַּבֵּק בְּמַחֲשָׁבוֹת זַכּוֹת צַחוֹת וּקְדוֹשׁוֹת בַּעֲבוֹדָתְךָ בֶּאֱמֶת בְּהַשָּׂגָתְךָ וּבְתוֹרָתְךָ.
only to cling to thoughts that are pure, clear and holy, in serving You in truth, in understanding You and in Your Torah.
What it means
only to cling to pure, clear and holy thoughts, in serving You in truth, in understanding You and in Your Torah.
The second line said no thought can take hold of God. Here the prayer asks for holy thoughts "in understanding You." The Guide for the Perplexed (Rambam's book of Jewish thought) says all agree that minds cannot grasp God, and only He grasps what He is.
Rambam, Guide for the Perplexed 1:59הַט לִבִּי אֶל עֵדְוֹתֶיךָ, וְתֵן לִי לֵב טָהוֹר לְעָבְדְּךָ בֶּאֱמֶת.
Turn my heart toward Your testimonies, and give me a pure heart to serve You in truth.
What it means
"Turn my heart to Your testimonies" is Tehillim 119:36, which goes on: "and not to gain." Radak: do not let my heart be eager to collect money. "A pure heart" is close to Tehillim 51:12, where David says "create." Rashi there: so that I do not stumble again.
Tehillim 119:36 and 51:12, with Radak and Rashiוּמִמְּצֻלוֹת יָם תּוֹצִיאֵנִי לְאוֹר גָּדוֹל חִישׁ קַל מְהֵרָה, תְּשׁוּעַת יְיָ כְּהֶרֶף עַיִן,
And from the depths of the sea take me out to a great light, right away, swiftly, soon: the rescue of Hashem like the blink of an eye,
What it means
A Midrash (Bereishit Rabbah, an ancient collection of the rabbis' teachings on Bereishit) has Rabbi Shmuel bar Nachman teach a prayer for each time the day changes. Evening: "May it be Your will to take me out from darkness to light." Morning: "I thank You that You took me out from darkness to light," the same first words as Modeh Ani.
Bereishit Rabbah 68:9לֵאוֹר בְּאוֹר הַחַיִּים כָּל יְמֵי הֱיוֹתִי עַל פְּנֵי הָאֲדָמָה.
to be lit by the light of life, all the days I exist on the face of the earth.
What it means
"The light of life" is in Iyov (Job) 33:30, in a speech about a person near the pit who prays, is accepted, and says "I sinned." Metzudat David (Rabbi David Altschuler, Poland, 1700s) reads that light as the life of the World to Come. This prayer asks for it here too: "all the days I am on the face of the earth."
Iyov 33:27-30, with Metzudat Davidוְאֶזְכֶּה לְחַדֵּשׁ נְעוּרַי הַיָּמִים שֶׁעָבְרוּ בַּחֹשֶׁךְ, לְהַחֲזִירָם אֶל הַקְּדֻשָּׁה,
And may I be worthy to renew my youth, the days that passed in darkness, to bring them back to holiness,
What it means
And may I be worthy to renew my youth, the days that passed in darkness, and bring them back to holiness,
וְתִהְיֶה יְצִיאָתִי מִן הָעוֹלָם כְּבִיאָתִי בְּלֹא חֵטְא.
And may my departure from the world be like my arrival, without sin.
What it means
This line repeats a passage of the Gemara (the rabbis' discussions, about 1,500 years old). Rabbi Yochanan (a leading sage of the Land of Israel, 200s) reads the blessing "blessed are you in your coming in and in your going out" to mean: may you leave the world as you entered it. As your entry was without sin, so may your exit be.
Bava Metzia 107a; Devarim 28:6וְאֶזְכֶּה לַחֲזוֹת בְּנֹעַם יְיָ וּלְבַקֵּר בְּהֵיכָלוֹ, כֻּלּוֹ אֹמֵר כָּבוֹד, אָמֵן נֶצַח סֶלָה וָעֶד.
And may I be worthy to gaze at the pleasantness of Hashem and visit His Temple, all of it saying: Glory. Amen, forever, always, for all time.
What it means
And may I be worthy to gaze at the pleasantness of Hashem and to visit His Temple, all of it saying: Glory. Amen, forever, always, for all time.
"To gaze at the pleasantness of Hashem and visit His Temple" is Tehillim 27:4, from L'David Hashem Ori. "All of it saying glory" ends Tehillim 29:9. Rashi there reads the Temple as the one that will be built, where everyone will praise Him.
Tehillim 27:4 and 29:9, with Rashi