Telling her why isn't something you should have to tell her.
When you moved in, if the agreement was that part of the deal included your husband and you being responsible for something like mowing the lawn, I think you'd have to either decide if you want to do the work yourself, hire someone to do it, or else tell the woman your husband won't be there and you can't keep up your end of the "work-around-the yard" arrangement.
If you husband has just been fixing things himself, rather than call the landlady, and she just likes that - that's something different again (of course, she'll be surprised if/when you call her for a repair issue, but you could tell her then).
I don't happen to know what laws there are that may say a landlord has a right to know exactly who is living in his property; or how it works if things change; so, there's the possibility that "technically" you're supposed to let the landlady know about any changes (but, again, I don't know the law on that). Even if you had to tell her, though, all you need to say is, "We're separated, and he isn't living here now." If she were to ask what happened all you have to say is, "I'd rather not get into it right now." If you have a copy of your rental agreement/lease you could see what it says on it, and maybe call an attorney to ask if whether you're required by law to tell the landlady at all. I believe free legal services are available, especially for landlord/tenant matters. The website for your court system may have a free legal information line listed under something like "resources". Or, an attorney may answer your question over the phone.