Since you give absolutely no specs for the device you're using, its hard to say much about it.
But, a typical laser diode (like a CD or DVD laser) will behave very much like an LED when it is biased below its "lasing threshold". Above threshold it will suddenly have very low equivalent resistance (even lower than an LED) and additional current driven through it will be very efficiently converted into optical power.
Right up until either the device overheats, either from excess IV in the electrical domain, or from excess optical loss (usually at the facets), which can quickly turn the device into a not-very-useful bit of exotic semiconductor.
In addition to providing careful current control, as other answers have mentioned, a power supply circuit for one of these devices also needs to provide overvoltage and overcurrent protection (often in the form of a crowbar circuit) in case the input voltage were to surge upwards, and ESD protection because an ESD event can also easily damage or destroy a laser diode.
The degree of sophistication needed to achieve the required protection depends a lot on how much power the laser is capable of producing and how much current is needed to drive it. Note for example, that the driver circuit you linked, designed for a 1 W laser, is extreme overkill in terms of power produced and probably ineffective in terms of protection offered, if used with the laser you linked (appears to be a 5 mW 650 nm laser in another Amazon listing) which needs only about 1/200 the power.
But if the part you bought is described is a "5 V laser module", that likely implies there's some kind of control circuit already built in. The threshold voltage for the laser diode itself is likely somewhere in the 2 - 3 V range. In that case you may only need to connect a 5 V supply to turn the laser on. But without any datasheet its not possible to say for sure that doing this won't just blow out the laser, and you'd still be wise to protect this input from voltage overshoots (like when turning on) and ESD.
Also be aware that a 5 mW laser is enough to cause eye damage if not respected.