A layer (.lyr) file is an ESRI file format that not intended (by ESRI) to be used outside of ESRI product (not an open format).
it has mainly two important functions:
- It contains a link or a pointer to the location of the Data.
- It contains information on how to render or draw the Data.
Note that the .lyr doesn't contain any data.
To use it in QGIS you may have a look the SLYR plugin (it seem that the import from .style is free while other functionality (like importing whole MXD or .lyr) are not).
Another option is to add the raw data to QGIS and style it yourself (more time consuming and if you need the exact same style you must know how it's supposed to look like)