Welcome to TeX.SE!
BibTeX basically works by reading the .bib and handled using the .bst-file. Bib(La)TeX then creates a LaTeX file (with a .bbl extention) based on the .bst files instructions, commonly consisting of a thebibliography environment with a lot of \bibitems. The .bst-file for example tells BibTeX that the journal entry should be included in the thebibliography environment.
If then there is this undefined control sequence \mnras in the journal entry, then LaTeX is going to complain when the .bbl file is included.
As the NASA ADS FAQ says:
Please note that our BibTeX entries make use of the AASTeX macro definitions for the most popular astronomical journal. If you're not using the AASTeX package then you will need to include these macros in your LaTeX source.
In short all you have to do is download the contents this file, save it as aas_macros.sty and then write \usepackage{aas_macros} in your preamble.
In the "general" case of fixing this for empirical errors I suspect the easiest course of action would be, as @mico suggests in the comments, to just define the undefined control sequence to what you want it to show e.g. \providecommand\mnras{Monthly Notices of the Royal Astronomical Society}. (\providecommand is like a \newcommand but just doesn't do anything if the command already exists).
if you want the actual backslash to show, as the question title suggests, then you'd escape said backslashes using e.g. \textbackslash; for example replace \mnras with \textbackslash mnras. In this case it would show literally "\mnras" in your references text.
If you want to redefine how backslash behaves in bibtex, you're in for quite an adventure, and you'd might want to dedicate a day or two to do so.
\providecommand\mnras{Monthly Notices of the Royal Astronomical Society}to the preamble and recompile the document? More generally, do the undefined-macro-name error messages vanish if you (a) download and save the fileaas_macros.styfrom http://adsabs.harvard.edu/abs_doc/aas_macros.sty and (b) provide the instruction\usepackage{aas_macros}in the preamble? – Mico Jan 01 '19 at 01:42