This has inspired me to finally write to you and ask if there is a possibility for me to join the painting course this spring.

inspired 是动词 inspire 的过去分词形式。inspire 由前缀 in-（表示"进入"）和词根 spire（"呼吸"）组成，原意是"注入气息"，引申为"鼓舞、激励"的意思。在这里 inspired 表示"被激励"的状态。