It states (Shemos 18:20) התורות ואת החקים את אתהם והזהרתם. The pasuk is speaking about teaching Torah, but Targum Yonason says the pasuk refers to tefillah. He translates the pasuk as follows: דיצלון צלותא ית להון ותהודע כנישתהון בבית, "Teach them the tefillos they should say in the beis knesses." So, the pasuk refers to both Torah and tefillah. This is because Torah and tefillah work together.
שלמה לך האלף writes, "People call this day Yom HaMeyuchas because it sits between two holy days, rosh chodesh and shloshes yemei hagbalah." Aruch HaShulchan (494:7) writes, "We don't say tachanun on rosh chodesh. The three days before Shavuos are shloshes yemei hagbalah, festive days. Therefore, the day in between also becomes part of the celebration. Furthermore, on this day, Moshe instructed the nation to be holy. It is therefore called Yom HaMeyuchas. Another reason is that the day of the week of Yom HaMeyuchas will also be the following Yom Kippur."
We can explain these words with a mashal: An eagle crawled along the seashore. Someone called out, "Eagle, Eagle! Why are you crawling? You have wings. Raise your wings and fly."
Shavuos is the yahrzeit of Dovid HaMelech. The meforshim say that Shavuos is also Dovid HaMelech's birthday since tzaddikim are niftar on their birth date. This is one of the reasons we read Megillas Rus on Shavuos. The final words of Megillas Rus are דוד את הוליד וישי, "Yishai gave birth to Dovid." We want to read this pasuk on the day Dovid was born.
The Ben Ish Chai (Bamidbar 6) recommends saying Tehillim on Shavuos. He writes, "It is important to say Tehillim on Shavuos because Dovid HaMelech a'h was niftar on this day, and the Tehillim said on this day will be more accepted in heaven. Everyone should say the entire Tehillim on Shavuos." The Kaf HaChaim (494:34) writes, "It is ideal to learn Tehillim on Shavuos because Dovid HaMelech was niftar on this day (as stated in the Yerushalmi)... Tehillim that one recites on this day will be answered."
There is an interesting contrast: Dovid HaMelech said about himself (Tehillim 109:4), ואני תפילה, "I am tefillah." Similarly, the Gemara (Brachos 7:) tells us that רות is called רות because ותשבחות בשירות הוא ברוך להקדוש ריווהו, she had a descendant, Dovid HaMelech, who made Hashem satisfied and content with his songs and praises." Yet, Dovid was niftar on Shavuos, on a day of Torah. The Chidushei HaRim zt'l says this is to tell us that Torah and tefillah are connected and dependent on one another. One can't succeed in Torah without tefillah, and one can't succeed in tefillah without Torah.
The Beis Aharon of Karlin zt'l once told his chassidim from Poland (who were with the Beis Aharon for yom tov and were about to return), "Polish Yidden love learning Torah, and so do we. The difference is that we say (Megillah 28:), צילותא בעי שמעתתא, which can be translated as, 'Learning Torah needs tefillah'. This is because it is impossible to succeed in Torah without tefillah. And it is impossible to daven properly without Torah."
