Consulting a dictionary on the meaning of 貽 makes the meaning of the expression rather clear. I would say “give (others cause for) great laughter”, “become laughingstock”. Yet I have trouble understanding 方。 The only function I can guess is that of a classifier. But there are two problems.
Firstly, it occurs in somewhat literary Chinese, which, I am given to understand, does not use classifiers.
清·李汝珍《鏡花緣》第17回:“才女才說學士大夫論及反切尚切瞠目無語,何況我們不過略知皮毛,豈敢亂談,貽笑大方!
On the other hand the use of 我 perhaps shows it is not really classical, but a hybrid literary Chinese?
A lesser problem is that none of my Pleco dictionaries (PLC, TL, CC, UNI) list 方 as a classifier for 笑。 Google search of 一方笑 doesn’t remotely support it as a classifier.