The column Correspondents shows either the sender or the recipients. In order to make the distinction in which is shown, the arrow indicator is added: the arrow shows it's an outgoing message and the addresses under Correspondents are the recipients of the message you sent.

In your Sent folder this doesn't make too much sense, since all the messages are outgoing messages, so you might consider changing the column to the Recipients columns instead of the Correspondents column.

These arrows were never there before. So why now?

Because otherwise the distinction between sender and recipients in the correspondents columns isn't clear. It is a new feature. At first they started with two arrows, one to the left for incoming messages and one to the right for outgoing messages, but that way you had to look very closely at the arrow and it may also have cause problems on RTL-languages, so the devs decided to only show an arrow in the Correspondents column for outgoing messages to indicate that the names are recipients.

Thank you Onno.

The arrows came with the Correspondents column. Before that, the message direction was implicit in the type of folder. If you have an Inbox and a Sent folder and leave your messages in there, the arrows are unnecessary. Inbox will show incoming messages, Sent will show outgoing messages. You would have had the From column in Inbox and the Recipient column in Sent. But all your messages to your family are in your Sent folder, and their messages to you are in your Inbox.

If you file messages in what seems to me to be a logical way, incoming and outgoing collected together according to context, subject or correspondent, then the Correspondents column is a lot more efficient than having both From and Recipient, and the arrow helps one to understand which way the message went. So you file all those messages in the "family" folder, and use Correspondents, which will automatically show who you sent to, or who sent to you. You don't need to see your own address in there, do you?

If you don't "get" Correspondents, or have no use for it, swap it for one of the old To or From columns and the arrows will go too.

There was quite some discussion about how to indicate incoming/outgoing messages, see bug 1266126. If you don't want to use the Correspondents column for all folders, you can set the pref mail.threadpane.use_correspondents to false.

Yep, I recall the discussion about left/right vs up/down. Your comment about RTL languages is interesting.