In the Burns poem, "To A Mouse," when the speaker refers to the mouse as a fellow mortal, he is saying that we are all mortal, we all die. Mice and men alike live their short time on earth then die. The speaker is acknowledging that in this respect, mice and men are equal. We put in our time and then we perish.
To be mortal is to end our time with death. This is a fundamental connection men have with mice in the poem. Between birth and death mice and men alike suffer hardships (like having our homes destroyed, specifically) and having our best laid plans go awry (in general).
Burns uses local language (his Scottish dialict) to present local people, common people with common problems. Burns brought poetry to the masses, to the commoners. Rather than deal with society's large issues, he deals with everyday existence. We live our lives and then we die, sharing our state of existence with the natural world.
In my opinion, the speaker is acknowledging the idea that all of nature is in some way connected. If you think about it, this is really the theme of the whole poem.
In this poem, the speaker is apologizing to the mouse for having broken up her home with his plow. He is doing this because he acknowledges how important the home was to her and how much trouble he has now caused her. By acknowledging this, he is implying that her life is important even though she is an animal.
Burns also talks in this poem about how "men" like mice, can have their worlds upset by outside forces. (The famous line about the best laid plans...) Once again, this acknowledges the kinship between the two life forms.
By calling the mouse a fellow mortal, the speaker is acknowledging that they are similar. They both have to try to make their way in a difficult world.