You can adjust the thermo profile or primer concentration of your reaction to reduce primer dimers, or just cut the proper band out of your gel, there are plenty of kits for it:

https://www.addgene.org/protocols/gel-purification/

https://www.qiagen.com/us/spotlight-pages/ias/automated-qpcr-workflow/purification/qiaquick-gel-extraction-kit/

If you want to separate it for cloning purposes, gel purification would be suitable as suggested by Peter. Run the PCR result on 2 - 2.5% agarose, then cut it and purified the specific DNA using available kits.

In my opinion, the simplest way is by increasing the annealing temperature (I often start from +5 degree above Tm).

This also may help:

1. Decrease the primer concentration

2. Increase initial denaturation time

3. Try to dilute the template and use the high purity DNA

4. Try to use PCR enhancer (DMSO works best for me)

5. Use high quality enzyme, Hot-Start Taq always work better for me

and if all the suggestion are not working, maybe it is better to evaluate your PCR primers for high possibility of hairpin, self- and hetero-dimer. I always use OligoAnalyzer from IDT to evaluate the primer specificity prior to order.

If the self- and hetero-dimer of your primers are high (especially at the 3' end) it might be better to re-design your primer.

Good luck

The primer dimer is a very common and usual circumstance that occurs in PCR. It can be resolved through the following techniques-

1. Substitute the value of annealing temperature during PCR

2.modify the timing and temperature of template denaturation.

3. use DMSO to enhance PCR.

4. check your primers design, use OligoAnalyzer 3.1 (http://eu.idtdna.com/calc/analyzer) for checking your primers.

5. optimize your reaction such that when your reaction finishes, there are very few primers left to form the dimers. You can try to scale back the primer concentration in your reaction.

6. try using 10pmole  concentration, else you can use a high-fidelity DNA polymerase, e.g. Pfu. 

7. Dilute your primers into working concentration. Best to use in 0.2 uM concentration. For this take 10ul Stock primers + 90 ul Dnase/Rnase free water. Add 0.5 ul of each primer into your PCR reaction mixture.

See if these points help in overcoming your problem in your current research.