The following is quoted from What is Mac Lane Missing? by Mathias.
The hypothesis that they [measurable cardinals] exist, or the hypothesis that in some inner model there are measurable cardinals may be construed as saying that in certain circumstances the direct limit of well-founded structures is well founded. Other large cardinal axioms may also be interpreted as assertions of this general kind.
Mathias thus argues that measurable cardinals (or even stronger large cardinals) are quite natural, since on the previous page he defines set theory to be the study of well-foundedness. How exactly are measurable cardinals related to direct limit of well-founded structures? I suppose the direct limit in iterated ultrapowers is not what Mathias is talking about here.
I am mainly interested because the consistency of measurable cardinals is usually justified in extrinsic ways (iterated ultrapower is a beautiful theory, relation to determinacy, etc.) while Mathias seems to give some intrinsic evidence.