What chemical conversions are involved , and what's the name for the process, when the muscles use lactate as an energy source?
I see you've already got an answer in the comments, so this is rather a collection (with more proper formulation) of comments. As is known, lactate is produced by muscles (generally) during exercise due to absence of oxygen. However, the lactate produced can be used again in following two ways:


Conversion of lactate to glucose in the liver via a process known as Cori's cycle. Basically, the lactate produced from the muscles is released into the bloodstream, through which it reaches liver. There, it is converted back to glucose and again released into the bloodstream, through which muscles can use it again as fuel. The basic cycle looks like:


Conversion of lactate to pyruvate is the one you are concerned about. However, this process does not have a special name, because this reaction can be catalyzed both in forward and reverse direction. In the absence of oxygen, an enzyme lactate dehydrogenase convertes pyruvate into lactate, consuming 2 H+ during the process. However, when the envrionment is rich in oxygen, or the concentration of lactate is high, the same enzyme converts lactate into pyruvate, producing 2 H+ in the process. The reaction can be shown as: $$\ce{CH_3COCOO^- + NADH + H^+ \leftrightharpoons CH_3CH(OH)COO^- + NAD^+}$$

Thus, when the concentration of lactate increases, the enzyme lactate dehydrogenase catalyzes the (above) reaction in the reverse direction. After this, the pyruvate produced can be used in the TCA cycle.


PS: there have been studies which now suggest that indeed lactate, not glucose, is the primary energy source of neurons. As per the lactate-shuttle hypothesis, the glial cells neighboruing neurons convert glucose into lactate, which is then used by the neurons. This fits with the observation that the extracellular fluid immediately surrounding the neurons is richer in lactate as compared with blood and cerebrospinal fluid, as shown in microdialysis studies. These studies suggest that glucose is not the actual primary energy source of neurons.

EDIT: As you see, the first reaction in both the cases is $\ce{lactate \rightarrow pyruvate}$. However, the main difference arises after this reaction. In muscle cells, this pyruvate is used in Krebs cycle, for which the first enzyme is pyruvate dehydrogenase converting pyruvate to acetyl-coA. However, in gluconeogenesis i.e. formation of new glucose molecule (a part of Cori's cycle), the first step is conversion of pyruvate to oxaloacetate, and is catalyzed by the enzyme pyruvate carboxylase. Thus, although the first step in both processes is the same, the end product is quite different.

References:


Lactic Acid - Wikipedia
Lactate Dehydrogenase - Wikipedia
Cori's Cycle - Wikipedia