There are two main sources of the idea that truth has degrees.
One is objective idealism, as explained under coherence theory of truth.
The other arises because many predicates are essentially vague. When does a heap of sand become large? It seems plausible that a 100-grain heap is not large, and that a heap which is not large can never become large by acquiring just one extra grain.
It seems to follow that a small heap which has grains added one at a time can never become large. (This is one version of the paradox known as that of the heap, the sorites, or the bald man; being invented in ancient Greece and only revived quite recently.)
The paradox (like many) is harder to answer than appears at first, and one answer takes the form of letting truth have degrees.
R M Sainsbury, Paradoxes (1988), ch. 2