请多多包涵 is a formal, courtesy expression. It says: something undesirable has happened, please accept my apology and thank you for your understanding. The basic meaning of 包涵 is to take in and to contain. Metaphorically, it means accepting something with grace.
Dear Dim Sum hasn’t been posting as frequently as I’d like, and probably won’t be for the next few weeks. I recently bought a home that needs work and suddenly got more things to think about than I am used to. Thank you for your understanding and patience.
(listen:) thank you for understanding
• 请: please. Pronunciation: qǐng (3)
• 多: a lot, more. Pronunciation: duō (1)
• 包: bun, package. Pronunciation: bāo (1)
• 涵: contain, include. Pronunciation: hán (1)
For other expressions related to apology, see this post.