Subgraph Isomophism Problem - Color Coding Technique - Proof Sketch
The idea is to repeat the proof of Theorem 6.1, only during the recursive construction in the second paragraph, we use the tree decomposition of $H$ rather than $H$ itself. We pick a vertex in the tree decomposition, which correspond to up to $t+1$ vertices of $H$, and split $H$ into two parts $H',H''$ accordingly, applying the algorithm recursively. Since we are dealing with groups of up to $t+1$ vertices at a time, we have to calculate the color sets for sets of $t+1$ vertices rather than for single vertices.

There are many resources online for this general approach, dynamic programming for graphs of bounded treewidth. You can consult Section 2.4 of Hajiaghayi's thesis or a presentation by Marx.