Approximately one year ago, I submitted some of my research work to arXiv.
Recently, I got an email from Lambert Academic Publishing saying that they are interested in publishing this work and stating that their services are free of charge for authors.
I don't have much idea about different journals and publications. It would be great if someone can project the benefits of moving to publish in such academic publications. As arXiv already provides open access, is there any need of publishing somewhere else also?
As you become established as a scholar, you will receive more and more of such invitations, most of which are either scams or trash publications attempting to trick you into giving them content. A small fraction, however, will actually come from legitimate colleagues who are inviting you to a special issue, editorial, or similar such opportunity.
Thus, when receiving an invitation, you should thus begin by evaluating the quality of the venue and the person who is inviting you, just as you would if you were choosing where to submit without an invitation. Be aware, however, that some disreputable publications will lie about the identity of people involved, and that some careless academics may have been tricked into lending their name to disreputable publications.
In the specific case of Lambert Academic Publishing, I will note that they appear on Beale's list as well as a number of other highly critical analyses, including on Wikipedia and this site. You can make up your own mind about their credibility, but I wouldn't recommend soiling your C.V. by publication with them.
Finally, to answer your question about arXiv vs. other publications: being in arXiv is fine for being accessible and citable. If your work is good, however, I would recommend submitting to a reputable peer-reviewed journal. If you are published in such, it will improve the reputation, notability, and value of your publication.