Chazal (Mechilta Shemos 20) say that Hashem appeared to the Jewish nation at matan Torah רחמים מלא כזקן, like an elderly person full of compassion. If we think about the word מלא, full, it reminds us of a cup filled with liquid. Every time someone lifts the cup or touches it, some liquid spills out. Similarly, on Shavuos, the day Hashem has a lot of compassion and He wants to give us a lot of Torah, every tefillah we say will shake the Kisei HaKavod, and Torah and bounty will pour down onto us.
Reb Menachem Rikanti zt'l yearned and pined to know Torah. For this purpose, he fasted many days and davened with all his heart. But he was born with a weak mind and wasn't succeeding in Torah. Nevertheless, he never gave up. He fasted, davened, and tried with all his might. When he was in his eighties, he dreamed that an elderly person gave him a cup of water and told him to drink it, which he did. In the morning, he was a new person. His heart was open to understanding Torah. In just a year or two afterwards, he completed his holy sefarim, renowned throughout Klal Yisrael.
The Chazon Ish zt'l taught that we shouldn’t lose hope in a child's ability to grow in Torah. The Chazon Ish explained that when his grandmothers lit the Shabbos lecht, they davened that their descendants should succeed in Torah. The moment these tefillos are answered, everything will turn around. The Chazon Ish said that a person can cross the street; when he began crossing the street, he had a weak mind, and when he reached the other side of the road, he had a wise and sharp mind, able to understand Torah. This is because that was the moment the tefillos were answered.
Don't be stingy with your words when you daven for success in Torah (and for all other areas in ruchniyos). Ask for a lot because Hashem can give you whatever you ask for. It states (Tehillim 81:11), ואמלאהו פיך הרחב, and Rashi translates these words to mean, "Open your mouth wide, ask for all your heart's desires, ואמלאהו, and I will grant all your requests." The Gemara (Brachos 50.) says this pasuk refers to success in Torah. So, open your mouth wide, ask for a lot, ואמלאהו, and Hashem will give it to you.
Sefer Chasidim (131) states, "If someone davens for something that will increase Hashem's praise — such as success in Torah or anything else that is Hashem's will — and he pours out his heart in tefillah, Hakadosh Baruch Hu will answer his tefillos, even if he doesn't have good deeds."
The Gemara (Bava Metzia 59.) states, "From the time the Beis HaMikdash was destroyed, the gates of heaven are closed." Reb Yisrael Salanter zt'l said this is solely when requesting worldly matters. But when one davens for ruchniyos, the gates are open.
The Sfas Emes ('וכ ה"ד ג"תרמ) writes, "It states באמת יקראהו ...'ה קרוב, 'Hashem is near...to those who call to Him with truth', and 'truth' is Torah, as it states (Avodah Zarah 4:), אלא אמת אין תורה. Therefore, every tefillah said from the midst of Torah is accepted. As Chazal (Brachos 31a) say, one should begin tefillah after studying Torah. Certainly, on this day of Shavuos, which the entire day is a day of Torah, it is an eis ratzon for tefillah. In Megilas Rus [which we read on Shavuos], it tells about the birth of Dovid, and he was the root of tefillah."
