Whichever miner mines the block that a transaction is included in must collect at least enough in transaction fees to cover the energy spent on mining. Since the block size (and thus number of transactions that can be recorded per block) is fixed, this implies that any increase in the global price of energy will raise Bitcoin transaction fees (all things being equal). This assumes that the transaction rate is always at the maximum possible; if the transaction rate is too low and blocks are not being filled, the effect is even more pronounced as there are fewer transactions over which the energy cost can be recuperated (last I checked, Bitcoin is already operating at its maximum transaction rate, but someone may want to correct me on that).

The reason it impacts coin prices is that higher transaction fees make Bitcoin less desirable as a medium of exchange and reduces demand. To put it another way, let's say I am paid in Bitcoin. If transaction fees increase, I will need to demand a higher nominal salary, at least enough to cover the transaction fees incurred when I spend my salary. Likewise, any merchants that accept Bitcoin payments will raise their prices, to cover the fees they will have to pay. Yet the same work is being done and the same goods are being sold; hence, inflation, or in other words, the value of Bitcoin has decreased.