How did Buffy beat The First? I know she got wounded in the battle, but I don't remember The First dying. Did it die/get defeated when Spike blew up the Hellmouth?

The First just can't be killed 'cos it's evil in its primeval form and has no consistency on its own, but only in that of the persons it possesses (e.g. Caleb) oder influences (e.g. Spike, Andrew).
The First's plan in Season 7 was to rise an army of UberVamps beneath the Hellmouth and unleash it through its dichtung under Sunnydale's High to take over the world.
Buffy and the Potentials (later become Slayers on their own) defeat that army in "Chosen" and Spike closes the Hellmouth once and for all, sacrificing himself for Buffy and mankind.
Von the way, I suggest Du to watch S7 in its entirety, 'cos it's really worth viewing (although not being the best season of the serie).

Thanks for reading.

Yes i think

I recall the scythe being involved and the First saying something darkly humorous... Funny, I just watched that episode again a few months ago.

But the First can't die... because it's the First Evil. No, I don't think the First did die, but Buffy/Spike thwarted its plans.

Du ask alot of buffy and Angel – Jäger der Finsternis Fragen did Du even watch these shows? Du should watch them