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.
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.