빠지다 in this sentence means "fall out;come out"
빠져 나가다 means "get out, escape, evade"
그런데 배가 너무 불러서 도저히 빠져 나갈수가없었지요.
However, because his stomach is too full, he just could not escape from there.
인제 is same to 이제 means "now"
됐군 is a short form of "되었구나"
composed of
되(다)+었+구나
되다 means "become"
었 is a past tense particle
구나 is an ending partcle with a nuance of exclamation.
인제 됐군 means "Now, the issue was sovled"
ㄹ/을수있었던거예요 is short form of ㄹ/을 수 있었던 것이에요.
ㄹ/을 수 있(다)+었+던+것+이(다)+어요
ㄹ/을 수 있(다) means "be able to do something"
었 is a past tense particle
던 is incomplete past particle
것 or 거 means "a thing"
이다 means "is"
어요 is an ending paticle
빠져 나올 수 있었던 거예요. has same meaning with 빠져 나올 수 있었어요.
But, it is emphasizing the result.
사흘을 굶은 여우는 다시 홀쭉해져서 울타리를 빠져 나올 수 있었던 거예요.
The fox starved for three days, and he became so hollow as before that he could escape out of the fence.