As Margana says in a comment above, Hamlet in Hamlet (in act 3 scene 1) expresses a somewhat similar idea:
Who would fardels [burdens] bear/ To grunt and sweat under a weary life,/ But that the dread of something after death,/ The undiscovered country from whose bourn/ No traveller returns, puzzles the will,/ And makes us rather bear those ills we have,/ Than fly to others that we know not of?/ Thus conscience doth make cowards of us all;/ And thus the native hue of resolution/ Is sicklied o'er with the pale cast of thought,/ And enterprises of great pith and moment/ With this regard their currents turn awry,/ And lose the name of action.
And to similar effect, the conversation between two murderers in Richard III (act 1, scene 4):
1 Murderer. Where's thy conscience now?
2 Murderer. O, in the Duke of Gloucesters purse.
1 Murderer. When hee opens his purse to give us our reward, thy Conscience flyes out.
2 Murderer. 'Tis no matter, let it goe: There's few or none will entertaine it.
1 Murderer. What if it come to thee againe?
2 Murderer. I'll not meddle with it, it makes a man a Coward: A man cannot steal, but it accuseth him ; A man cannot Sweare, but it Checkes him ; A man cannot lye with his Neighbours Wife, but it detects him. 'Tis a blushing shamefac'd spirit, that mutinies in a mans bosome ; It filles a man full of Obstacles.
In both of these instances, the notion is that "conscience"—the sense of right and wrong—may restrain our willingness to commit wrongful acts as easily as we otherwise might, and indeed may sometimes actually prevent us from acting in a harmful or sinful way. And both Hamlet and the second murderer seem to characterize the fear of punishment after death as a form of cowardice—and so it is, if judged by its effects, for it stays the hand that might other wise strike confidently and remorselessly.
The originator of the wording "Reflection makes men cowards," however, appears to be William Hazlitt, in Characteristics: In the Manner of Rochefoucault's Maxims (1823), who very conveniently (though briefly) explains what he means by the phrase:
Reflection makes men cowards. There is no object that can be put in competition with life, unless it is viewed through the medium of passion, and we are hurried away by the impulse of the moment.
What Hazlitt means is that in a moment of high emotion, we may temporarily forget the danger to ourselves of taking some bold, heroic action, and merely act on behalf of anger, a burning desire for revenge or justice, or some other passion. But as soon as rationality sets up shop and begins weighing the probable costs and benefits of various actions, the preciousness of one's own life dominates one's thoughts and one becomes exceedingly reluctant to risk parting with it on any but the most favorable odds. Hazlitt is using reflection in the sense of definition 6 or (even more likely) definition 7 below, from Merriam-Webster's Eleventh Collegiate Dictionary (2003):
reflection n (14c) 1 : an instance of reflecting; esp : the return of light or sound waves from a surface 2 : the production of an image by or as if by a mirror 3 a : the action of bending or folding back b : a reflected part : FOLD 4 : something produced by reflection: as a : an image given back by a reflecting surface b : an effect produced by an influence [example omitted] 5 : an often obscure or indirect criticism : REPROACH [example omitted] 6 : a thought, idea, or opinion formed, or a remark mde as a result of meditation 7 : consideration of some subject matter, idea, or purpose ...
Considering a situation from a realistic perspective, Hazlitt suggests, makes us proceed with maximum caution and minimum endangerment to ourselves. In effect, it makes us cowards.