This very nice question was recently asked on the Reddit forum r/askphilosophy. Here is the (reposted) answer.
Russell’s reasoning is given in the passages immediately preceding that quote. Russell wants to argue that, in the case of symmetrical relations, we can understand them as oneplace predicates if we wish. This is because we can, using oneplace relations, preserve the usual inferences involving symmetrical relations. For example:

A and B both have property G. [premise]

B and C both have property G. [premise]

Thus, A and C both have property G. [from 12]
On a theory involving only oneplace relations, this inference can be preserved. This is because A and B, and C, too, all have the same property. So there is no problem of relating A’s color to B’s color, and then to C’s color. They all have the same color. So one could at least see how the oneplace relations story could go for symmetric relations.
The case of asymmetric relations is entirely different. Russell’s view is that, if we try to eliminate all relations in favor of oneplace relations, then we cannot preserve ordinary inferences involving asymmetric relations. Take Russell’s example:

A is greater than B. [premise]

B is greater than C. [premise]

So, A is greater than C. [from 12]
Russell says about this sort of example:
Take for example “A is greater than B”. It is obvious that “A is greater than B” does not consist in A and B having a common predicate, for if it did it would require that B should also be greater than A. It is also obvious that it does not consist merely in their having different predicates, because if A has a different predicate from B, B has a different predicate from A, so that in either case, whether of sameness or difference of predicate, you get a symmetrical relation.
Russell is outlining two strategies for accounting for this clearly valid inference, assuming we only have oneplace relations. We can say that A being greater than B consists in them having (a) the same property, or (b) different oneplace properties.
Strategy (a) fails. A having the property being greater than B does not imply that B has this same property (indeed, B cannot have this property). So we have to introduce distinct properties for A and B. That is, we must pursue strategy (b). But then we have the problem of relating these oneplace relations together.
This problem cannot be overcome. There is no way to get from the first two claims to the third (absent, perhaps, some modal logical principles about tallness). Just consider how that argument would look if we only have oneplace relations:

A has the property being greater than B.

B has the property being greater than C.

So, A has the property being greater than C.
This argument gets the form:

F(A)

G(B)

H(A)?
And it won’t help to add oneplace relations either, so that A being taller than B consists in their having different predicates. Just consider:

A has the property being greater than B and B has the property being smaller than A.

B has the property being greater than C and C has the property being smaller than B.

So, A has the property being greater than C.
But (3) doesn’t follow still. The argument has the form:

F(A) and P(B)

G(B) and Q(C)

H(A)?
So, in short, if we eliminate all relations in favor of oneplace relations, then there is no practicable way to make inferences involving the asymmetric or transitive properties of relations.