In order to see texting activity, you either need access to the phone in question, or access to paper or online billing documents. If you're wanting to look at your own texting activity, this is a very simple process, although it'll be a little more tricky if you're wanting to check someone else's account.
If you can't remember who you've been texting, you can check by looking through your text message history, which will display the phone number at the top of each message.
Alternatively, you can contact your network provider to request a statement that lists the phone usage. This can either be done via a paper statement, or alternatively, by logging into your online account on your network provider's website.
How to view someone else's text message history
If you're looking for someone else's text message history, it's going to be a little more tricky. If you have access to their phone, you could try to browse through past text messages, although this doesn't provide you with a detailed list of activity.
The other option is to find the account details of the person in question, and log into their account to look at their statements for yourself. Of course, to do this you'd need to know their account password which could be difficult to obtain, but this really is your only option at this point.
A word of warning
Although being able to see someone's text message history could be something you feel you need to do, it's likely to be discovered by the person in question, and that could lead to issues in your relationship. Mobile phone account details are private for a reason, and I'd recommend that you simply ask the person who they've been texting, rather than going behind their back to find out.