What do the % variances mean on a Bray curtis dissimilarity plot? 
I'm new to microbiome data analysis but what do the percentages on the Axis mean? This plot is comparing two groups (SPF n=8 and ABX n= 4 mice). I can't figure out what these mean. I thought the axis were arbitrary and set to just show the distances as best as possible.
Thank you!

They are not variances in the usual sense because you used a Bray-Curtis dissimilarity matrix rather than a variance-covariance matrix. But the idea is similar.
If you do not already, it is worth understanding a closely-related technique called principal components analysis. See this this post and its answers to get a sense of what it is about.
While PCoA (aka MDS) is not identical to PCA, it does extract eigenvalues and eigenvectors from a given matrix. Each percentage in your plot is the ratio of the eigenvalue of that basis vector to sum of the eigenvalues for all basis vectors for a given choice of orthogonal basis. Such a basis is an orthogonalized eigenbasis, which is obtained from the Bray-Curtis matrix you provided.
Your plot suggests that the first principal component contributes about 50 % of the total score weights (what would otherwise be variance), and visual inspection suggests that is separates the two groups of points in you data. The second principal component contributes much less to the total weight, and does not appear to separate the two groups of data.
