Intuitively, it is to indicate degree of 'truthiness'. Certainly false {F} is less truthy than either both true and false {T,F} and neither true nor false {}, which are in turn less truthy than certainly true {T}. The Hasse diagram indicates that there is no relationship between {} and {T,F} in terms of level of truthiness.

More mathematically, you can think of the Hasse diagram as a lattice, with the meet and join operations being AND and OR. For example, we are used to thinking of the expression

  p AND q

as being true if p is true and q is true. But what if we have four values: {}, {T}, {F} and {T,F}? The Hasse diagram tells us how to interpret expression like this - p AND q is the meet (greatest lower bound) of p and q, and p OR q is the join (least upper bound) of p and q. So for example,

  {F} AND {T}   = {F}
  {F} OR  {T}   = {T}

  {F} AND {}    = {F}
  {F} AND {T,F} = {F}
  {F} OR  {}    = { }
  {F} OR  {T,F} = {T,F}

  {T} OR  {T,F} = {T}
  {T} OR  {}    = {T}

  { } OR  {T,F} = {T} // I think..!
  { } AND {T,F} = {F}
  { } OR  { }   = { }
  { } AND { }   = { }

The Hasse diagram in the article makes sure that expressions like this agree with our intuition in the cases where we have intuition for what the result should be, and give us a way to interpret expressions consistently when our intuitions fail us.

We're used to seeing Hasse diagrams for subsets of a set S used to indicate inclusion, in which case you would have {} < {F}, but you can have a valid poset structure that's not based on inclusion.