I don't pretend to have anything more than a superficial understanding of condensed mathematics, but Scholze's lecture notes (on condensed mathematics and analytic geometry) are so clearly written that you can get some sense of the main idea just from studying the first few pages.
The starting point is the observation that the traditional way of endowing something with both a topology and an algebraic structure has some shortcomings. The simplest example is that topological abelian groups do not form an abelian category. We know from long experience with category theory that an excellent indication of whether you have a "good" definition of an object is that the category of all your objects (and the maps between them) has nice properties, and topological abelian groups fail to have some of those nice properties. In condensed mathematics, the category of topological abelian groups is replaced by the category of condensed abelian groups, which is an abelian category.
Although the goal is easy enough to state and motivate, the method of achieving it was initially not obvious even to Scholze, one of the architects (along with Clausen) of condensed mathematics. A key role is played by a category that might seem unpromising at first glance: the category $\mathcal{S}$ of profinite sets, a.k.a. totally disconnected compact Hausdorff spaces (or Stone spaces), with finite jointly surjective families of maps as covers. Most people's first impression of totally disconnected spaces is that they're weird, and they have some trouble even thinking of examples other than the discrete topology. However, categorically speaking, $\mathcal{S}$ has some nice properties. A condensed abelian group, roughly speaking, a special kind of (contravariant) functor from $\mathcal{S}$ to the category of abelian groups (somewhat more precisely, it is a sheaf of abelian groups on $\mathcal{S}$). That is, the way a topological structure is imposed on abelian groups is not in the classical way (i.e., by taking a set and defining a group operation and a topology on it in isolation), but by taking certain functors from this funny-looking topological category $\mathcal{S}$ to your algebraic category.
There's nothing about this story that is peculiar to abelian groups; by replacing "abelian group" with "set" or "ring" you get condensed sets and condensed rings and so forth.
The benefit of this shift from classical structures to condensed structures is not just aesthetic. One of the nicest applications is that it leads to new proofs of certain classical results in algebraic geometry. A longstanding puzzle (if you want to call it that) is that certain theorems about complex varieties that "feel algebraic" seem to be provable only via "transcendental methods"; i.e., by invoking analysis in a seemingly essential way. Condensed mathematics provides new proofs of some of these classical theorems that are more algebraic. See Condensed Mathematics and Complex Geometry by Clausen and Scholze for more details.
