Is there a known mechanism for mass-energy distorting spacetime?

Classical general relativity

If "mechanism" means "accurate description in terms of something more fundamental," then classical general relativity doesn't provide that. In classical GR, the equation that connects spacetime curvature to mass-energy is the most fundamental thing, as explained in Dale's answer.

However, classical GR can be motivated in a relatively simple way, and for the purpose of giving students some inspiring insight, the motivation might be good enough.

Here's the idea: One of the pillars of our current understanding of nature is the action principle, which may be loosely translated as saying that influences go both ways. If $A$ influences the behavior of $B$, then $B$ must also influence the behavior of $A$.

For example, if an electromagnetic field can influence the motion of a charged object (that's the Lorentz force equation), then the motion of a charged object must also influence the electromagnetic field (that's Maxwell's equations). Mathematically, the Lorentz force equation and Maxwell's equations can both be derived from a single action (the integral of a lagrangian), and this ensures that the charge$\to$field and field$\to$charge influences are related to each other in a special way. Most importantly for this question, both influences exist.

It’s clear how curved spacetime geodesics make straight line paths looked curved in other frames but not how the curvature is created by the presence of mass-energy.

Spacetime geometry is described by the metric field. If students accept that spacetime geometry (the metric field) influences the motion of material objects, then the action principle says that the influence must go the other way, too: material objects must influence the metric field — that is, they must influence the geometry of spacetime.

For students who are comfortable with derivatives, the idea behind the action principle can be introduced like this: If $f(x,y)$ is a single function of $x$ and $y$, then the functions $$ g(x,y) := \frac{\partial}{\partial x} f(x,y) \hskip2cm h(x,y) := \frac{\partial}{\partial y} f(x,y) \tag{1} $$ are related to each other in a special way. Specifically, they satisfy $$ \frac{\partial}{\partial y} g(x,y) = \frac{\partial}{\partial x} h(x,y). \tag{2} $$ In words: "if $g$ depends on $y$, then $h$ must also depend on $x$ in a related way (and conversely)." This is analogous to the action principle: influences must go both ways.

The real action principle for general relativity involves a single function $S(\text{metric},\text{matter})$, called the action, that depends on the metric field and on other entities that are traditionally called matter (which in this context includes the electromagnetic field). Schematically, the equation that describes how the metric field influences matter may be written $$ \frac{\partial}{\partial\,\text{matter}} S(\text{metric},\text{matter}) = 0, \tag{3} $$ and the equation that describes how matter influences the metric field (the equation shown in Dale's answer) may be written $$ \frac{\partial}{\partial\,\text{metric}} S(\text{metric},\text{matter}) = 0. \tag{4} $$ The trivial identity $$ \frac{\partial}{\partial\,\text{matter}} \left( \frac{\partial}{\partial\,\text{metric}} S(\text{metric},\text{matter}) \right) = \frac{\partial}{\partial\,\text{metric}} \left( \frac{\partial}{\partial\,\text{matter}} S(\text{metric},\text{matter}) \right) \tag{5} $$ is analogous to the identity (2). It says that if the metric field can influence the behavior of matter in equation (3), then matter must also influence the behavior of the metric field in equation (4) in a specially-related way. Details aside, the important message is that matter must distort the geometry of spacetime.

What principles dictate the precise form of the action? To address that, we have Lovelock's theorem: the part that describes the way matter distorts spacetime (Einstein's field equation) is essentially uniquely determined by (a) general covariance, (b) the assumption that spacetime is four-dimensional and locally like Minkowski spacetime, and (c) a technical condition on the number of derivatives in the lagrangian. But again, the action principle itself already says that the influence must go both ways; these additional conditions just secure the details.

A ray of hope for a better answer...

Over the past few decades, hints have been accumulating that gravity (the distortion of spacetime by matter) may be some kind of thermodynamic phenomenon and that spacetime as we know it is just an approximation to something deeper.

This started with the observation that the laws of black hole mechanics look just like the laws of thermodynamics (superficially, at least), but with an entropy that scales with area instead of with volume. Then came Hawking and his derivation of black hole radiation, which agreed perfectly with what the thermodynamic analogy suggested. Then came an accelerating flood of additional insights, like Jacobson's early paper Thermodynamics of Spacetime: The Einstein Equation of State and many more recent papers like Gravitational Dynamics From Entanglement "Thermodynamics".

Most of the recent work is related to the surprising realization that GR emerges naturally from certain lower-dimensional quantum field systems with very strong interactions. This is called gauge/gravity duality, and it embodies the holographic principle. The very strong interactions make these lower-dimensional systems difficult to analyze directly, which is probably the main reason why this connection went unnoticed for so long. Maybe someday it will be understood well enough to offer a more satisfying answer to this Physics SE question.

In physics, “how” something happens is described by the equations that govern it. So in this case the explanation about how mass-energy curves spacetime is given by Einstein’s field equations.

This is a set of ten coupled differential equations which relate specific changes in the curvature of spacetime at a given location to the amount of mass-energy at that location. Usually, this equation is written in tensor form because otherwise it would take many pages to write down. In tensor form, and natural units, it has the deceptively simple appearance: $$R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu} -\Lambda g_{\mu\nu}=8\pi T_{\mu\nu}$$

The “source” term $T_{\mu\nu}$ is called the stress-energy tensor. It contains energy density, which is primarily due to mass, momentum density, pressure, and shear stress, all of which contribute to the spacetime curvature in a highly non-trivial way.

Unfortunately, you may find this explanation unsatisfying. Often, we prefer explanations which can be easily and succinctly expressed in English (or your native language). But our natural vocabulary simply doesn’t have the words to convey the physical relationship that is expressed by the Einstein field equations. That doesn’t make the explanation any less valid, just less palatable.