Remember that mass and energy are equivalent, right?
So when an object moves, it has kinetic energy, in addition to its rest mass. Therefore, it kind of makes sense (not really) to add its rest mass and its kinetic energy to arrive at the concept of “relativistic mass”.
One problem with this concept is that motion is relative. Look at the person next to you. Is he moving? Perhaps not. Relative to you, he is at rest. But relative to the solar system reference frame, he and you are both moving at the considerable speed of about 30 km/s. So which value should we use to calculate your relativistic mass? Is it 0 km/s or 30 km/s? It all depends on the observer.
However, there is one situation in which relativistic mass does contribute to the actual mass of an object: internal motion. So for instance, if you heat an object, that means that its atoms and molecules are moving, bouncing, vibrating about slightly faster. And yes, the energy you invest into heating the object will contribute to its total mass, due to the increase of this internal kinetic energy.
Last but not least, this is about any speed less than the speed of light. The speed of light itself is unattainable.