"Ederim" comes from the root "et", meaning "to make happen".
Remember "içerim" (I drink) from the previous lessons? It is constructed as iç+er+im, with the iç being the root, -er making simple present and -im making first person singular.
Same here. Et+er+im (I make happen). Note that the hard "t" became a soft "d". This is a rule/lesson you'll learn later.
Now, back to the word at hand.
"Rica" means "request".
"Rica ederim", thus means "I [hereby] make [my] request happen". In more understandable terms, it's "I request".
What is it exactly that you're requesting? It's that the person you've helped not make a big deal out of your assistance. So it's short version of "I request [that you don't make a big deal of this]. It was my pleasure".
"rica" means "request". "rica ederim" can be translated as "I request", literally. But, this is the shortened version of "I request you not to mention it". So, it is a phrase for "no problem", "you are welcome", etc. And it is both polite and common in daily language. So you can use it everywhere.
not necessarily. "etmek" (to do), "yapmak" (to make) and "olmak" (to become) are verbs which are used to derive more verbs from nominal words. They are some sort of "helping verbs" in these situations.
- araştırma yapmak: to research
- kaybetmek (kayıp+etmek): to lose
- yemek yapmak: to cook
- hissetmek (his+etmek): to feel etc...
so, "etmek" does not make things formal.
rica etmek: to request
and that is the short version of "I request you not to mention it". You simply say "I request..." and the other person understands what you request without you saying it.
in case of "teşekkürler" and "teşekkür ederim", it is very similar to "thanks" and "thank you" in english.
I hope that clears some things up. :)