What is this unit attached to my house, and what do I replace it with?
That's telephone.  It's not a transformer.  It's a terminal block. 

Call up the phone company and ask them to have someone come out and show you where your point of demarcation is.  That's where the wires stop being their responsibility and start being your responsibility (unless you have an in-house service plan as part of your bill).  It looks like an obsolete one of those.  If so, you can't mess with "their" half of it (the side going off to the pole). 

If it is downstream of your point of demarcation, you can do as you please, at peril of breaking your landline phone service. (and Internet if it comes via phone line).