As briefly mentioned in Beginner Tips and Official Guide, Travel is an useful method to obtain Basic Himes every 24 Race Point (1hr travelled = 1 Race Point) accumulated for an attribute as well as some Experience for the travelled hime. Aside from getting more himes, brushes are also obtainable based on the attribute and tier of the travelled hime that reaches the 24 Race Points. The higher tier the hime travelled is, the higher the chance to obtaining the listed brushes.
Note - If your Race Points exceed 24, the counter still resets to 0 and the exceed amount won't be carried over. e.g You are at 23 Race Points and set your hime to travel for 8hr, your counter will be 0/24 once the hime returned.

List of obtainable Brushes from Travel:

Spear Recovery Axe Magic Sword Bow
Brush of Life
Brush of Life - Mid
Brush of Strength
Brush of Strength - Mid
Brush of Defense
Brush of Defense - Mid
Brush of Speed
Brush of Speed - Mid
Brush of Training