This is really common, so don't panic: you are not abnormal!
The reason behind it is low oestrogen levels. Oestrogen acts as a mild antidepressant and generally makes us cheerier. With less of it around we can feel very emotionally vulnerable and weepy, even though our brain tells us that there is no real reason for our feelings. There are several things you can do to make yourself less vulnerable.
Firstly you could consider taking a phyto-estrogen, which is gentle oestrogen found in plants, which can make you feel more emotionally robust without any of the side effects associated with a strong oestrogen like HRT. Check out Menopause Support for this one option.
Secondly you could consider taking Hypericum, if you are not on any other medication. This is a mild plant-based antidepressant that many women find very supportive during the menopause. I suggest using the tincture (liquid) format if you go for this one, as it works faster than the tablets.
Thirdly, exercise makes you produce more natural antidepressants in your own body, so although it's usually the last thing you feel like doing when your low mood strikes, a 10-20 minute walk or quick jog around the block or even 5 minutes with a skipping rope will make you feel a lot better. If you get used to doing a small amount of exercise most days - just 10-20 minutes, nothing drastic - then you will be better protected against low mood.
You may also find Mood Essence or Confidence Essence helpful, and these can be taken alongside any other medication or supplement.