Learn Swedish: Jag ser många fåglar i skogen. - I see many birds in the forest.

Because fåglar (birds) are things you can count. We use många for countable nouns, just like we use "many" in English. You would use mycket for uncountable things, like mycket vatten (a lot of water).

The word for a bird is fågel. Why is the plural fåglar instead of "fågelar"?

When a Swedish noun ends in -el (like fågel), it usually drops the e before adding the plural ending to make it easier to pronounce. So, fågel drops the e and adds the plural ending -ar to become fåglar.

Why is the word skogen and not just skog?

Skog just means "forest", but the English translation is "in THE forest". In Swedish, you usually express "the" by adding an ending to the noun. Adding -en to skog gives you skogen (the forest).

Would the verb ser change if the sentence was "we see" instead of "I see"?

No, Swedish verbs do not change based on who is doing the action. The verb stays exactly the same, so you would say jag ser (I see), vi ser (we see), and de ser (they see).