Trustworthiness is a moral value considered to be a virtue

Virtue

Virtue is moral excellence. A virtue is a positive trait or quality subjectively deemed to be morally excellent and thus is valued as a foundation of principle and good moral being....

. A trustworthy person is someone in whom you can place your trust and rest assured that the trust will not be betrayed. A person can prove their trustworthiness by fulfilling an assigned responsibility

Responsibility

Responsibility may refer to:* Collective responsibility** Cabinet collective responsibility, a constitutional Convention in Governments using the Westminster System* Corporate responsibility** Corporate social responsibility...

 - and as an extension of that, not to let down expectation

Expectation

In the case of uncertainty, expectation is what is considered the most likely to happen. An expectation, which is a belief that is centered on the future, may or may not be realistic. A less advantageous result gives rise to the emotion of disappointment. If something happens that is not at all...

s. The responsibility can be either material, such as delivering a mail package on time, or it can be non-material such as keeping an important secret to themselves.
A trustworthy person is someone that you can tell your worries and secrets to and know they won't repeat them without your permission.