。
Thank you for inviting me to dinner.
Literal
Dinner [to-に] [humble-お]-invitation give-[polite-past-ました], thank-you-very-much.
A polished thank-you. お招き is the humble/polite nominal form of 招く ('to invite, beckon') — noun use with the honorific お- prefix. 下さり is a slightly formal variant of くださり (the masu-stem of くださる, the honorific 'give'), so お招き下さり literally means 'having given me the invitation.' The stylistic chain is characteristic of polite thank-you speech at weddings, dinners, or formal gatherings.