Hajimemashite is one of my favourite Kagamine Len songs. The reason: he actually hits some low notes. The song is by oORaikaOo, who also made the song My Girl. If anyone wonders, the picture is from the PV for the Kagamine Rin song Be Mine!.
Copy & pasted from the video description to explain the meaning of some of the words:
"Yoroshiku onigai shimasu" is a first introductory greeting towards someone who is going to take care of you. (eg. husbands to wives and vice versa, worker to their boss) "Hajimemashite" is a first introductory greeting towards acquaintances (eg. you friend introduced you to her cousin), but has the a stem that literally means "starting from now" so there is a double meaning to the word in the song.
Translated by kiathian.