Can we use
got in this case? Is it right and usual?
after that meeting I got enthusiastic about traveling
In your example, "got" is appropriate to use.
I got enthusiastic
I got tired
I got hungry
has more a feeling of a quick change, in contrast to
I became enthusiastic
I became tired
I became hungry
The common phrase
I got it!
is often used to signify grabbing/grasping something quickly, like an idea, in a "light bulb moment"
I got enthusiastic about travelling.
The above sentence is correct grammatically.
You can use the verb
get as a linking verb in the sense of becoming; it links the subject (l) of the sentence with the adjective enthusiastic. Some examples are:
He is getting ready.
It's getting late.
I got tired when I was working in the garden. It's got dark now.