Hebrews 1.3 also says of the Son that he sustains all things by his powerful word. I think there are at least two senses in which this is true: first, in an ontological sense, Jesus or the λόγος of God sustains the universe in existence from moment to moment; second, in an existential sense, Jesus' word keeps our lives ordered, structured, meaningful.
David B. Hart's The Experience of God does a good job of explaining the kind of metaphysical reasoning which motivates the belief that God sustains things in being from moment to moment. The idea is that the world is ontologically impoverished: things exist which cannot of themselves explain the fact that they exist. Consider the case of myself: I exist, but nothing about me explains why I exist. I exist because the organic matter which composes my body is arranged in such a way as to produce a living human body, but nothing about me explains why that is so. It is clear that this organic matter could exist just fine in a different form -- you could chop up my body into small pieces, in which case the matter would still exist but I would not. But not only I but more or less everything is like this. More or less everything is living on borrowed existence, so to speak, since no material thing can explain why it exists of itself. To simplify greatly, the conclusion Hart and the classical theistic tradition draws is this: therefore there is God, who is a fullness and infinity of being, who as a matter of definition has existence of himself, and he gives existence to everything else. Burning candles are ultimately lit by a flame burning of itself; so also contingent reality owes its existence to God who has being of himself.
Stăniloae too regularly speaks of the rationality and knowability of the universe as requiring the existence of a kind of transcendental Mind or Reason of which it is its thought. We consider that the rationality of the cosmos attests to the fact that the cosmos is the product of a rational being, since rationality, as an aspect of reality which is destined to be known, has no explanation apart from a conscious Reason which knows it from the time it creates it or even before that time, and knows it continually so long as that same Reason preserves its being (The Experience of God: Orthodox Dogmatic Theology, p. 2). Because the world has such a structure as allows it to be known -- which is a critical presupposition of the possibility of knowledge in the first place -- we infer that it is itself in a way that manifested thought or knowledge of a greater Mind.
These of course are very abbreviated statements of longer arguments. In any case, the point is this: the world is of such a nature as to require a sustaining cause, a cause which has existence and the power of creation entirely of itself. The author of Hebrews identifies Jesus (who is the Son of God and the λόγος of God) with that cause.
But Jesus also sustains our lives in an existential manner through his word. To know that surely I am with you always, even to the very end of the age (Mt 28.20), or that your Father has been pleased to give you the kingdom (Luke 12.32) gives us strength to keep on going in the face of the senselessness of much of life. To know that the promise is for you and your children (Acts 2.39) and that the Father is more than happy to give the Holy Spirit to those who ask (Luke 11.13) gives you confidence to approach God, even in the darkest times of your sin. And as I've posted recently, the knowledge that we have peace with God through Jesus Christ (Rom 5.1) can help us to approach adversity and suffering as opportunities to develop ourselves in the likeness of Jesus Christ. These are ways in which the meaningfulness of our lives can be sustained by Jesus' word.