だ is the casual form of です, it means “is” or “it is”.
してる is a shortened form of している, which is the て form of する (to do) + いる (exists) = している (or しています if you want to be formal). It means “is doing”.
Example:
サッカーをする = to play soccer
サッカーをしている = is playing soccer
愛 (あい) is a noun that means “love”. To me, saying 愛だ is almost like saying “it’s love” instead of “to love”. That would be 愛する (noun + suru = to do the noun, so “to do love” = to love).
好き(すき) is a な adjective that means “likeable”. When you say 好きだ you are saying “[it’s] likeable”. That’s why you can say 好きだ to mean “you are likeable” = “I like you”.
愛している means “I love you”, (literally, “I am loving you”) but as @plantron mentioned, it’s super extra, like saying, “I REALLY, REALLY LOVE YOU” and is used more in songs, romance novels, dramas, etc. If you say it to a person it comes off as particularly strong.
Misa from Japanese Ammo breaks this down precisely in this video:
