If $m=0$, then your sextic is $x^6+L$.  For this to have a real root, you need $L\leq 0$, in which case those real roots are the real sixth roots of $-L$.

We may now assume $m\neq 0$.  Then take $K:=L/m^6$.  After dividing your sextic by $m^6$, and taking $y:=x/m$, you have a new (simpler) sextic $f(y):=y^6-3y^5+5y^4-5y^3+3y^2-y+K$ with only one parameter.  This new sextic has real roots exactly when the old one did (just scaled by $m$).

The derivative of $f$ has only a single real root, at $y=1/2$.  Thus, the minimum of $f$ occurs at $f(1/2)=-\frac{9}{64}+K$.  In order for there to be a real root, you need $K\leq \frac{9}{64}$.  If you run 

> Manipulate[
 Plot[y^6 - 3 y^5 + 5 y^4 - 5 y^3 + 3 y^2 - y + K == 0, {y, 0, 
   1}], {K, 0, 9/64}]

in Mathematica, you can visually see how the two roots behave, as $K$ varies.

Because the minimum is at $y=1/2$, we make the linear shift $z:=y-1/2$.  Then, the sextic transforms to $$z^6+\frac{5}{4}z^4+\frac{3}{16}z^2+\left(K-\frac{9}{64}\right).$$

Notice that this is really a cubic in $z^2$, with a negative constant term.  Multiplying through by $64$, to remove the denominator, replacing $4z^2$ with $w$, and $64K-9$ with $J$, we are really just looking for the (unique) nonnegative real root of
$$
w^3+5w^2+3w+J
$$
when $J\leq 0$.

Thus, it is not surprising after all that Mathematica found solutions to the original sextic, since cubics are always solvable using radicals.

Running the code

> Solve[w^3 + 5 w^2 + 3 w + J == 0, w]

you get three expressions involving radicals.  When $J=0$, only one of these expressions is nonnegative (and is equal to $0$).  That expression is the one that will ultimately give you your nonnegative real root, because that expression is continuous as a function in $J\leq 0$.

Now that you have your solution for $w$ in terms of $J$, you can transform everything back to your original system (if you like).