Is ב״ה Baruch HaShem or Bezrat HaShem?
Why did The Rebbe begin every letter with it?
ב״ה can have two meanings in this context. It can mean B'ezras HaShem and Baruch HaShem.
The Lubavitcher Rebbe, Rabbi Menachem M. Schneerson, of righteous memory, had the custom of starting off his letters with ב״ה, as can be seen in his thousands of published letters, and he encouraged others to start off their letters acknowledging G‑d in a similar fashion.
I've seen that B'H (with one ') means: Baruch Hashem and that B"H (with two ") means Be'ezras Hashem.
The Rebbe urged people to write B"H and to begin with B"H with all relevant matters. This can be found in מנחם משיב נפשי - Menachem Meishiv Nafshi 2:650, bringing Igros Kodesh, letter 1384.
“It is of upmost importance in today’s times to begin all relevant matters with the abbreviations of Beis Hei”
And in Likkutei Sichos vol. 6, p. 190:
“It is the custom of all Israel, and a custom of Israel is Torah, that upon writing even a mundane letter, they begin to write Beis Hei and the like.”
Please note that it is not acknowledged everywhere. Rav Soloveitchik says (Harerei Kedem B 124) that it is a newly created custom, and that our sages would never use it in any form on their letters.