Learn Hardcore Spanish (Spain): El ver la lluvia desde mi cuarto me relaja.

Because an infinitive can act like a noun in Spanish. Here, el ver la lluvia means seeing the rain or the act of seeing the rain. The article el helps make ver feel more noun-like.

Could I also say Ver la lluvia desde mi cuarto me relaja without el?

Yes. That is also grammatical and very common. Without el, the sentence sounds a bit more direct and natural in everyday speech. With el, it sounds slightly more formal or reflective. Both versions work.

Why use ver instead of viendo?

Because Spanish normally uses the infinitive when an action is the subject of a sentence. In this sentence, seeing the rain from my room is the subject, so ver is the normal choice.

Viendo la lluvia usually works more like while watching the rain or watching the rain, not as the main subject in this kind of structure.

Why is it me relaja and not me relajo?