Short answers, pulled from the story.
Interlacing in The Lord of the Rings is a narrative device where multiple storylines progress simultaneously and intersect, reflecting the confusing flow of events people perceive in daily life. This technique appears outside France as well, found in Ovid's Metamorphoses from ancient Rome and the Old English epic poem Beowulf.
John R. Holmes published a definitive study in 1975 focusing specifically on the interlace structure. He argued Tolkien used this medieval technique in a decidedly modern way, closer to Virginia Woolf and William Faulkner than to Thomas Malory or Chrétien de Troyes.
Tolkien structured the novel as six books, though publishers chose to print them as three volumes for practical reasons. Interlacing begins in earnest from book three when the Fellowship breaks apart and different groups pursue their own quests.
J.R.R. Tolkien delivered his famous lecture Beowulf: The Monsters and the Critics on the subject of Beowulf. He was an expert on Beowulf and studied Edmund Spenser's 1590 Faerie Queene which drew heavily from Italian epics like Ariosto's Orlando Furioso.
Emily Auger detects all three types of narrative interlace used by Tolkien in Alan Lee illustrations: structural interlace achronological order, stylistic interlace restatement themes, pictorial interlace characters environment depicted outer projection inner self. She lists over 50 scenes where various forms of interlacing occur throughout trilogy runtime.