Is "in layman's terms" a slang idiom? If it is, what would be the best way of saying it in an academic work.
I want to explain these terms in a layman terms.
I think what you're asking is "Is 'in layman's terms' formal enough for an academic paper?" The answer is yes, it is OK to use that expression in your paper. It is not informal or slang.
As Alan recommended in his comment 'in layperson's terms' is a good alternative if you want to make an effort to use gender-neutral terms. I personally think that layperson is a little awkward, but it can be important in some environments to choose more 'inclusive' words.
I would write "I will explain my conclusions in layman's terms." In my opinion, you should not say 'explain terms' if you are talking about certain technical words, you should say 'define terms'; I'm not sure everyone agrees with me about that. You could explain the 'terms' of a contract, but I think that is a different meaning from the one you intend to write.
If you think "layman" is related with the sexual connotation of the verb to "lay", you are overthinking the idiom. Layman just means non-professionals and it is used to distinguish them from professionals. Merriam-Webster explains its etymology which will help you understand why it is OK to use in any context.
Layman began its run in English as the open compound "lay man." In this context, "lay" is an adjective that can mean "belonging or relating to those not in holy orders," "not of the clergy," and "not ecclesiastical." The origins of "lay" and "layman" can be traced back through French and Late Latin to Greek laikos, meaning "of the people." Layman was originally used to distinguish between non-clerical people and the clergy, but it was soon also being used to distinguish non-professionals from professionals in a field (such as law or medicine).
You could rephrase your example sentence to:
I want to explain these terms in a succinct manner or in simple terms.
But I don't think "in a succinct manner" or "in simple terms" are better than "in layman's terms".