On negative oxidation states, in general
Although it's usually a topic that's covered relatively late in a chemistry education, negative oxidation states for transition metals[1] are actually quite alright. On the Wikipedia list of oxidation states, there are quite a number of negative oxidation states. Some textbooks have tables which only show positive oxidation states, but good authors will be careful to add a disclaimer along the lines of "only 'common' oxidation states are listed".
The key is in the ligand that the metal is paired with. In metal ions with positive oxidation states, the ligands that they pick up tend to be Lewis bases, like $\ce{H2O}$, $\ce{Cl-}$, $\ce{NH3}$, ... This should "make sense" intuitively, because the positively charged metal ion wants to gain electron density, and these ligands are happy to give electron density to it.
In $\ce{Mn(CO)5-}$, the ligand is carbon monoxide: a very poor Lewis base, but it does possess low-lying π* orbitals. In the parlance of coordination chemistry, CO is a poor σ-donor, but a great π-acceptor. This means it tends to seek out metal centres which are relatively electron-rich instead of electron-poor, and thus it makes sense that metals with a negative oxidation state are its preferred partners. [I'll defer a full explanation of the $\ce{M-CO}$ bonding to a proper inorganic textbook.]
It turns out that it's actually quite difficult to find examples of carbonyl complexes with positive oxidation states.[2] Nearly all carbonyl complexes have the metal in oxidation states of zero or lower: see, e.g. $\ce{[Cr(CO)6]}$, $\ce{[Fe(CO)4]^2-}$, $\ce{Ni(CO)4}$, and of course, $\ce{[Mn(CO)5]-}$. Another common feature is that all of these complexes obey the 18-electron rule.
On [HMn(CO)5]
So, back to the question of the oxidation state in $\ce{[HMn(CO)5]}$... IUPAC has the final say on oxidation states;[3,4] see also this writeup by Martin. Quoting from ref. 4,
Oxidation state equals the charge of an atom after its homonuclear bonds have been divided equally and heteronuclear bonds assigned to the bond partners according to Allen electronegativity [...]
Following this, we have to assign the electrons in the $\ce{Mn-H}$ bond to the hydrogen, since hydrogen is more electronegative than manganese on the Allen scale (3.30 and 2.75 respectively). There is also evidence that in metal carbonyl hydrides like these, the hydrogen bears a partial negative charge, although this argument is not without its caveats.[5] Therefore, the oxidation state of $\ce{Mn}$ should be +1. However, IUPAC offers a way out of this:
[...] except when the electronegative atom is bonded reversibly as a Lewis-acid ligand, in which case it does not obtain that bond’s electrons.
According to this, it is possible to have exceptions to the general algorithm for determining oxidation state in the previous quotation, and there is a case to be made for this exception being valid here: the electronegative atom ($\ce{H}$) is indeed bonded reversibly ($\mathrm pK_\mathrm a = 7.1$) as a Lewis acid ($\ce{H+}$). If we accept this line of argument, then $\ce{H}$ does not "obtain" the two electrons in the $\ce{Mn-H}$ bond, and the oxidation state of $\ce{Mn}$ returns to −1, which is perhaps more in line with our chemical intuition.
I leave it to the reader to draw a conclusion for themselves; but this is an excellent illustration of the most important lesson here: oxidation states have their limits, and should not always be taken as meaningful descriptors of the chemical bonding in a species, especially when ambiguity is present, since this makes their assignment essentially arbitrary. If we assign an oxidation state of −1 to hydrogen here, it doesn't magically make the complex any less acidic than it was before. And IUPAC are aware of it (ref. 4):
The applications of OS in chemistry are wide and deal with a cornucopia of chemical compounds and materials. It is therefore not surprising that, for some compounds, one value does not fit all uses, or that dedicated measurements or computations are needed to ascertain the actual OS. In those rare cases when the most convenient OS becomes a matter of choice, this fact must be clearly stated.
Refs. 3 and 4 make for excellent (and fairly accessible) reading on the topic, for those wishing to find out more about these edge cases. Both can be accessed without a subscription.
Notes and references
Negative oxidation states for non-transition metals exist (e.g. alkalides), but these are probably more "exotic" than organometallic complexes.
They do exist, for example $\ce{[Ir(CO)6]^3+}$, although the bonding situation is markedly different (as evidenced by, e.g., IR spectroscopy).
Karen, P.; McArdle, P.; Takats, J. Toward a comprehensive definition of oxidation state (IUPAC Technical Report). Pure Appl. Chem. 2014, 86 (6), 1017–1081. DOI: 10.1515/pac-2013-0505.
Karen, P.; McArdle, P.; Takats, J. Comprehensive definition of oxidation state (IUPAC Recommendations 2016). Pure Appl. Chem. 2016, 88 (8), 831–839. DOI: 10.1515/pac-2015-1204.
Sweany, R. L.; Owens, J. W. Evidence for a negative charge on hydrogen for some metal carbonyl hydrides. J. Organomet. Chem. 1983, 255 (3), 327–334. DOI: 10.1016/S0022-328X(00)99326-4.
