He's not saying 'bigly'. He's saying 'big-league'. He uses what you hear often enough that sometimes in faster speech it sounds like 'bigly'; when slower he enunciates it as 'big league' (phonetically it is easy to drop (in pronouncing)/miss (in hearing) the 'g' because the diphthong /ij/ ends in the jot semi-vowel and final stops in English are not released.
Also the reason you tend to hear 'bigly' is because he tends to use the adjective 'big league' in a place that you'd normally hear an adverb. So as non-wordish as 'bigly' sounds, you ear hears that instead of the actual word 'big league'.
Now that that's cleared up, the meaning of 'big league' is essentially 'really' or 'a lot'. It comes from a sports metaphor, the big leagues are important, the main stage, but the minor league or little league are for those trying to get into the big league. So 'big league' is a noun metaphorically meaning a very important environment. For example:
I submitted my article to some minor journals, but my advisor said I should really shoot for the big leagues
The CEO resigned because he had an opportunity at a big league investment company.
That second example is already stretching the meaning a little, using it now as a noun modifier.
Now even further, to actual usages:
"That will be proven out, big league"
"We are going to start winning, big league"
Now it sorta makes sense, sorta.
It's not usual in English to do such a long path, from noun to noun-modifier to adjective to adverb to sentence modifier (but there is an example right there 'really'), but if not for the loose play with expected grammar, the semantics is not a terribly far step for the meaning to be 'very much' or 'a lot' or 'really, to an extreme'.