The Kli Yakar explains that these three would distract the king from his service to G-d. Excess amounts of wealth would have him forget G-d, as the verse says earlier in the Devarim (8:13-14). Owning many horses would remove his trust in G-d; instead he would rely on his horses and chariots to save him from harm (the daily prayer of "Lamnatzei'ach speaks of the other nations relying on their military while Jews rely on G-d). Having many wives would lead to him sinning and leaning towards their requests rather than G-d's; that was precisely what happened to King Solomon.
If the king follows these three prohibitions, his kingdom will thrive. The Kli Yakar notes that this is hinted to in the Hebrew word for chair - "Kisei". Kuf stands for Kesef (money); Samech stands for sus (horse); and Alef stands for isha (woman).