Sich freuen is a reflexive verb which means the action is reflected back on the subject.
These verbs need a reflexive pronoun like mich "myself". Reflexive verbs are a lot more common in German than they are in English.
The literal translation of "Ich freue mich über das Wetter." is "I please myself about the weather." In this case, the reflexive pronoun, mich, is in the accusative case.
Let's have a look at another example: The reflexive form of "to wash" is "sich waschen". Here you would say "Ich wasche mich." "I was myself."
Sometimes the reflexive pronoun needs to go into the dative case, for example "Ich wasche mir die Hände "I wash my hands." In this particular case the direct object is part of the speaker. You will find that the accusative form is much more common, though.
You can use sich freuen when describing your feelings in a particular moment such as "Ich freue mich von dir zu hören." - "I am happy to hear from you."
Glücklich is more commonly used when talking about your general emotional state, e.g.:
"Ich bin glücklich über mein Leben." - "I am happy with my life."
I hope this helps. Let me know if you have any further questions!