This is the first sloka(verse) of Bhagavatham by Sri Vyasa Bhagavan. The general meaning of this is that the Supreme Lord, being the cause of everything, the creator, sustainer and destroyer or dissolutionist of the worlds, is all knowing and is directly or indirectly connected with everything(all pervasive). He is supremely independent. He imparts knowledge into the heart of the first created being, Lord Brahma(Hirnyagarbha or Prajapathi), through transcendental sound. The Supreme Lord makes the ephemeral world appear real, and it is capable of throwing even the most learned sages like Visvamitra into illusion. Krishna says in Gita “mama maaya durathyaya”. I therefore, meditate on Him, who lives eternally in His spiritual abode(Paramdhamam) forever free from the ephemeral and delusional world. I meditate on Him for He is the Absolute Truth.

The word by word meaning is
Satyam - Supreme Lord who exists in all times and space(final and eternal truth).
Param - Highest God beyond whom there is nobody or nothing.
Dheemahi - Let us pray or meditate.

Satyam param dhimahi is the divine truth present in the form of a sound(Primordial sound). It means let us meditate on that one supreme truth. What is TRUTH? The answer is described as “Yad asthi tat sathyam” “Yath sath tat sathyam”. This is how it is described in Chandogya Upanishad(a manthra). Indeed let us meditate on the highest, and the absolute TRUTH. Every one has to realize that the goal of life is to search for truth. It is from that truth we will find freedom from all the unrealities and ephemeral nature of life. It is important for us to keep in mind that our freedom depends upon the extent to which we are in touch with the truth. By connecting with the truth all uncertainties and insecurities will disappear into oblivion.

The Vedic knowledge is described as the breathing of the Supreme Lord, and activities of world begin on the basis of Vedic knowledge. Therefore the Lord says that when He endeavors or breathes, the material universes come into existence, and various activities develop afterwards. The Lord says in Bhagavad-gita, pranavah sarva-vedeṣu: "I am the syllable Om in all the Vedic mantras. "Vedic knowledge begins with the vibration of the transcendental sound Praṇava/Omkara(primordial sound). It is an intonation sound, it is a universal sound and it is an alphabetical sound.

Although the sound vibration of the transcendental qualities of the Lord is equally beneficial to all, for those who are liberated(siddhulu), it is a blissful sound. As described in Srimad-Bhagavatam, serious devotees, like Prahlada who no longer have any material desires, surrender fully to the lotus feet of the Lord, they always merge in the ocean of bliss by chanting and hearing the Lord's holy name. Devotees like Narada are seen always engaged in chanting the holy name of the Lord because by such chanting they are always externally and internally blissful. The mumukshus, (persons desiring to achieve Moksha), do not depend on the pleasures of the senses. Instead, they concentrate fully on becoming liberated by chanting the holy name of the Lord(nama sankeerthana). Karmishti people like to create something pleasing to their ears and hearts(upasanas) and although they sometimes like to chant or hear the glories of the Lord, they do not do with regularity nor with deep love for it.

There is no difference between the Lord and the knowledge about him. Therefore, hearing about Him(knowing about Him) means immediate contact with Him by the process of Vibration of the transcendental sound. Because of illusion or maya created by the Lord, the material(gross body) body is accepted as an actual fact(real). It is like a magician’s show where everything is very enjoyable or sad when he cuts the woman’s body into two or fear when he produces a snake from his cuff. The world is as complex as the magician and his knowledge of magic. People will realize the magician’s show as an illusion but they identify the life as real although it is exactly the same, only produced and organised by the Lord. The living beings under different categories of life become illusioned in different ways. Even in the most developed stage of human life, the same illusion prevails in many forms and shapes and divides the loving relation with the Lord and thereby divides the loving relation between people. By studying the subject matter of Srimad-Bhagavatam this false complexity of materialism is removed(just as King Parikshit when taught by Sukhabrahma), and real peace in the human heart and thereby in the society begins. This is the way to Kaivalyam. And so “Satyam Param Dheemahi”.