Shorter T1 in Tissues
Let's consider what affects T1. As you know, the T1 relaxation time refers to interaction (energy transfer) between the excited spins (hydrogen protons) and the "lattice" -- the molecules which make up the surrounding structure. To exhibit efficient T1 relaxation, there must be good "contact" of spins with the lattice. This contact takes the form of an oscillating magnetic field at or near the Larmor frequency (this is just the resonance condition of magnetic resonance). And that occurs as a result of the movement of the molecule (water or fat molecules, mostly) in which a spin resides. We call this movement molecular tumbling.
This graph shows that soft tissues have shorter T1 times. The movement, or tumbling is rotation, vibration and translation of the molecule in which the hydrogen proton resides. Water molecules, which are the source of most of the signal in MRI, can be found in three states which affect their movement or tumbling. They can be free to move (free water); bound to a macromolecule by a single bond and so rotation is still possible ("structured" water), or bound by two bonds, and not allowed to rotate ("bound" water). Free water tumbles very fast, and so the frequency of the oscillating magnetic field is higher than the Larmor frequency, and it is not efficient at T1 relaxation (it has long T1s). Another way of describing this is with correlation times (τc). Free water moves alot, and doesn't spend much time interacting with macromolecules and so τc is short.
Water in soft tissues transiently interacts with proteins and other macromolecules. A higher proportion of structured-water molecules produce an oscillating magnetic field due to their movement near the Larmor frequency, and so are efficient at T1 relaxation. Structured-water has a medium range of correlation times.
In solids, water molecules are tightly "bound"—very restricted (the correlation time is long)—and so the frequency of the oscillating magnetic field which they produce is lower than the Larmor frequency. This means the T1s in solids are longer. Lipids (fat) also have long correlation times and short T1s. (Higher or lower than the Larmor frequency doesn't matter, it's an oscillating magnetic field at or near the Larmor frequency which shortens T1.)
When we say "transfer of energy to the lattice" we are referring to the oscillating magnetic field described above. This energy is then dissipated in the thermal motion of the molecules. We usually think of net magnetisation vectors when discussing T1, but we must think about processes occurring to individual protons to answer this question correctly. Note: contrast agents also affect T1.
Can you see why Felix Bloch likened relaxation to friction? The agitation (by magnetic fields) precipitates the transfer of energy.