The phrasal verb "well up" means "to fill up with a liquid," and is often used in the context of tears in the tear ducts around your eyes, or some kind of strong emotion. You can find these definitions on the TFD idioms page:
well up (from something) and well up (out of something)
for a liquid to gush or pour up and away from something : Tears welled up out of the baby's eyes.
well up (v.) to rise to the edge of a container, ready to flow : Tears welled up in my eyes, but I did not cry. I could feel anger well up in me.
Also, in this context, tissues simply refers to facial tissues, which can be used to wipe away tears, or blow your nose, as one often needs to do after or during a good crying session:
So, tissues close by simply means "have a box of tissues handy and within reach". The writer could have just as easily said:
There’s nothing wrong with crying! If you feel yourself getting very emotional and ready to cry and you’re in a quiet and private place, go for it! You’ll feel so much better. Just make sure you have your handkerchief ready!