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Lifting straps are a legitimate training tool for weightlifting—they unquestionably have a purpose and should be used at certain times with certain exercises by every weightlifter out there. The trick is knowing when this is and when it isn’t.
 
My basic rules on when straps should be used:

• Snatch & Clean Pull Variations
• Snatch & Clean Deadlift Variations
• RDLs & SLDLs

Exceptions to these are when a lifter really needs extra grip strength work, in which case, as many sets as possible of these exercises should be done without straps. However, keep in mind that with pulls in particular, an insecure grip will change the way the body moves—pulls will slow down and positions may even change if the body senses the grip is not secure. This is the primary reason to use straps in pulls—most lifters would be able to hang on to the bar during pulls without them, but the reduced grip security will negatively impact the movement too much. It’s better, in consequence, to use straps for pulls and work on grip strength in other ways.
 
The most common question I get is whether or not to use straps with hang snatches or snatch complexes in which the bar needs to be lowered from a previous rep without being returned to the floor. Generally my answer is only if absolutely necessary, and I’d prefer in most cases that the bar be dropped and picked up again without straps than straps used to lower it. The ability of the lifter to hold onto the bar securely is equal in importance to the ability of the hips and legs to accelerate and elevate it, or the upper body to hold it overhead. You can’t make a lift if any of these things is lacking. Long term, you’ll get the best results by training in a way that keeps the development of all of these traits on pace with each other.

Read these articles about grip strength for weightlifting:
Grip Strength and Training for Weightlifting
Grip Strength Program for Weightlifting

Watch a video on how to use straps: